Spatial dispersion

Centrographic measures of spatial dispersion, such as the standard distance, provide a numerical value to summarize the radial scattering of a set of points around their centre of gravity or centroid. This paper develops a procedure to test for the statistical significance of differences in dispersion between two sets of phenomena intertwined in space. The significance test is implemented using a resampling randomization procedure based on the pooled locations from both sets to estimate the sampling distribution of their differences. Repeated thousands of times, that yields empirical frequency thresholds of the sampling distributions to assess the statistical significance of the observed differences. Case studies based on residential locations of lone-parent families and retired couples in the Quebec City Metropolitan Area illustrate the procedure. This paper shows how randomization procedures can be used to adapt classical tests to assess the statistical significance of differences between indices of spatial dispersion.

Les mesures centrographiques de dispersion spatiale, telles que la distance-type, fournissent une valeur numérique pour synthétiser la diffusion radiale d’un ensemble de points autour de leur centre de gravité ou barycentre. Cet article développe une procédure pour tester la signification statistique des différences de dispersion entre deux ensembles de phénomènes qui s’entrecroisent dans l’espace. Le test de signification utilise un procédé de rééchantillonnage par randomisation qui combine les localisations des deux ensembles pour estimer les distributions d’échantillonnage des différences. En répétant des milliers de fois, on obtient des seuils empiriques de fréquence dans les distributions d’échantillonnage afin d’établir la signification statistique des différences observées. Des études de cas fondées sur la localisation résidentielle des familles monoparentales et des couples retraités de la région métropolitaine de Québec illustrent la procédure. L’article montre comment des procédés de randomisation permettent d’adapter des tests classiques pour évaluer la signification statistique des différences entre les indices de dispersion spatiale.

It is well known that second-harmonic generation (SHG) from dipoles within the bulk vanishes in inversion symmetric semiconductor materials as a consequence of parity symmetry. Hence, SHG is then either ascribed in the form of surface dipole effects or bulk related electric quadrupole or magnetic dipole effects. By incorporating the redefined spatial dispersion into the simplified-bond-hyperpolarizability model (SBHM), we show for the first time that the SHG spatial dispersion contribution for an Si(001) and Si(111) facet orientation can be reformulated by a third rank tensor containing one independent parameter namely the complex SHG spatial dispersion hyperpolarizability. Our results show that certain unexplained rota-tional anisotropy (RA) SHG intensity features for different incoming light wavelengths or optical penetration depth can be well reproduced only if contribution from spatial dispersion is incorporated.

I describe three ways that the spatial properties of a wave propagation medium can cause dispersion, and propose that they should form the basics for correctly understanding and naming phenomena described as "spatial dispersion". In particular, I emphasise the specific spatial properties which generate the resulting dispersive -- i.e. spatially dispersive -- behaviour. The mechanisms are geometry, structure, and dynamics.

See also arxiv version at: https://arxiv.org/abs/1904.11957